Motivation. Isogenies with a given polarized abelian variety give moduli points in a Hecke orbit. In characteristic zero any Hecke orbit is dense in this moduli space. What can be said about the Zariski closure of a Hecke orbit in positive characteristic?

Result. In joint work with Ching-Li Chai we prove a conjecture (1995) that predicted the answer to this question. One of basic tools is the notion of foliations. The main focus of the talk will be on describing properties of these foliations, rather well-understood by now.